A quantum ghost coupled polynomially to a harmonic oscillator has unitary evolution and a stable vacuum because a conserved quantity possesses a positive discrete spectrum.
Quantum mechanics with a ghost: Counterexamples to spectral denseness
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Unitary Time Evolution and Vacuum for a Quantum Stable Ghost
A quantum ghost coupled polynomially to a harmonic oscillator has unitary evolution and a stable vacuum because a conserved quantity possesses a positive discrete spectrum.